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The controversy was ostensibly over what gets to be the "true quantity of motion", momentum or vis viva (kinetic energy), with Newton and Leibniz on the opposing sides. While there was some philosophical angle at first it quickly deteriorated into a sideshow to the priority dispute over the invention of calculus, which fueled it, with the substantive part getting lost in all the acrimony.

After 1743 Euler and D'Alambert portrayed the controversy as a debate about words, which became the consesus. Here is Mach's summary: "Investigations of Newton really proved that for free material systems not acted on by forces the Cartesian sum Σmv is a constant, and the investigations of Hyugens showed that also the sum Σmv^2 is a constant... The dispute raised by Leibniz rested, therefore on various misunderstandings. It lasted 57 years till the appearance of D'Alambert's Traite de Dynamique in 1743".

The philosophical side was only tangentially related to energy and momentum. Leibniz criticized mechanistic Cartesian philosophy for not explaining the "source of the vitality" of matter. Newton concurred, but for this very reason to him force should have remained a fundamental concept of mechanics, irreducible to masses and speeds. So he opposed the elevation of vis viva to a metaphysical status favored by Leibniz. More details can be found here.

The starting point of the controversy was Descartes defining "momentum" as mass times speed (not velocity) in the tradition of medieval impetus, and claiming that its total value is conserved. Leibniz gave an example with falling bodies demonstrating that Descartes's "momentum" is not conserved. By that time Huygens already established that the sum of masses times speeds squared is conserved in elastic collisions (he also gave a form of "Newton's" second law), so Leibniz declared that the "true quantity of motion" and called it vis viva. In the meantime Wallis gave the correct definition of momentum as mass times velocity, and stated the conservation law correctly. Newton incorporated that into Principia terming it, you guessed it, "quantity of motion". And so it began.

Partly, the semantic debate was inevitable since the modern definitions of basic mechanical notions weren't established yet. Partly, separate issues got entangled with the original controversy. In 1724 Paris Academy offered a prize for the "best" way of calculating collisions between absolutely hard bodies. Johann Bernoulli's submission stated that... there are no absolutely hard bodies, all collisions are elastic, and, by the way, vis viva is the true quantity of motion. In return Maclaurin suggested calling mv the "force of bodies", and applying Newton's laws to it. Bernoulli's submission was rejected because he rejected the Academy's premise, and Maclaurin won the prize. In 1728-29 there was a scuffle over whose analytic methods are "better" for mechanics, Newton's or Leibniz's.

The controversy was ostensibly over what gets to be the "true quantity of motion", momentum or vis viva (kinetic energy), with Newton and Leibniz on the opposing sides. While there was some philosophical angle at first, a "*skillful attack by Leibniz against an inadequate concept, $m|v|$, and its description of the world*", it quickly deteriorated into a sideshow to the priority dispute over the invention of calculus, which fueled it, with the substantive part getting lost in all the acrimony.

After 1743 Euler and D'Alambert portrayed the controversy as a debate about words, which became the consesus. Here is Mach's summary: "Investigations of Newton really proved that for free material systems not acted on by forces the Cartesian sum $\sum mv$ is a constant, and the investigations of Hyugens showed that also the sum $\sum mv^2$ is a constant... The dispute raised by Leibniz rested, therefore on various misunderstandings. It lasted 57 years till the appearance of D'Alambert's Traite de Dynamique in 1743". Some modern scholars question this conclusion however, pointing out that the controversy lingered on "through the remainder of the eighteenth century", that a crucial observation on the issue only appears in the second edition of Traite de Dynamique (1758), and was made earlier by Boscovich (1745). Namely, "vis viva is the measure of a force acting through a distance while momentum is the measure of a force acting through a time".

The philosophical side was only tangentially related to energy and momentum. Leibniz criticized mechanistic Cartesian philosophy for not explaining the "source of the vitality" of matter. Newton concurred, but for this very reason to him force should have remained a fundamental concept of mechanics, irreducible to masses and speeds. So he opposed the elevation of vis viva to a metaphysical status favored by Leibniz.

The starting point of the controversy was Descartes defining "momentum" as mass times speed (not velocity) in the tradition of medieval impetus, and claiming that its total value is conserved. Leibniz gave an example with falling bodies demonstrating that Cartesian "momentum" is not conserved. By that time Huygens already established that the sum of masses times speeds squared is conserved in elastic collisions (he also gave a form of "Newton's" second law), so Leibniz declared that the "true quantity of motion" and called it vis viva. In the meantime Wallis gave the correct description of what happens to velocities in elastic collisions, which is equivalent to "conservation of momentum" (he uses no such language, and there was no notion of vector at the time to define "momentum"). Newton incorporated that into Principia but continued to call Cartesian "momentum", you guessed it, "quantity of motion". And so it began.

Partly, the semantic debate was inevitable since the modern definitions of basic mechanical notions weren't established yet. Partly, separate issues got entangled with the original controversy. In 1724 Paris Academy offered a prize for the "best" way of calculating collisions between absolutely hard bodies. Johann Bernoulli's submission stated that... there are no absolutely hard bodies, all collisions are elastic, and by the way, vis viva is the true quantity of motion. In return Maclaurin suggested calling mv the "force of bodies", and applying Newton's laws to it. Bernoulli's submission was rejected because he rejected the Academy's premise, and Maclaurin won the prize. In 1728-29 there was a scuffle over whose analytic methods are "better" for mechanics, Newton's or Leibniz's.

The controversy was ostensibly over what gets to be the "true quantity of motion", momentum or vis viva (kinetic energy), with Newton and Leibniz on the opposing sides. While there was some philosophical angle at first it quickly deteriorated into a sideshow to the priority dispute over the invention of calculus, which fueled it, with the substantive part getting lost in all the acrimony.

After 1743 Euler and D'Alambert portrayed the controversy as a debate about words, which became the consesus. Here is Mach's summary: "Investigations of Newton really proved that for free material systems not acted on by forces the Cartesian sum Σmv is a constant, and the investigations of Hyugens showed that also the sum Σmv^2 is a constant... The dispute raised by Leibniz rested, therefore on various misunderstandings. It lasted 57 years till the appearance of D'Alambert's Traite de Dynamique in 1743".

The philosophical side was only tangentially related to energy and momentum. Leibniz criticized mechanistic Cartesian philosophy for not explaining the "source of the vitality" of matter. Newton concurred, but for this very reason to him force should have remained a fundamental concept of mechanics, irreducible to masses and speeds. So he opposed the elevation of vis viva to a metaphysical status favored by Leibniz. More details can be found here.

The starting point of the controversy was Descartes defining "momentum" as mass times speed (not velocity) in the tradition of medieval impetus, and claiming that its total value is conserved. Leibniz gave an example with falling bodies demonstrating that Descartes's "momentum" is not conserved. By that time Huygens already established that the sum of masses times speeds squared is conserved in elastic collisions (he also gave a form of "Newton's" second law), so Leibniz declared that the "true quantity of motion" and called it vis viva. In the meantime Wallis gave the correct definition of momentum as mass times velocity, and stated the conservation law correctly. Newton incorporated that into Principia terming it, you guessed it, "quantity of motion". And so it began.

Partly, the semantic debate was inevitable since the modern definitions of basic mechanical notions weren't established yet. Partly, separate issues got entangled with the original controversy. In 1724 Paris Academy offered a prize for the "best" way of calculating collisions between absolutely hard bodies. Johann Bernoulli's submission stated that... there are no absolutely hard bodies, all collisions are elastic, and, by the way, vis viva is the true quantity of motion. In return Maclaurin suggested calling mv the "force of bodies", and applying Newton's laws to it. Bernoulli's submission was rejected because he rejected the Academy's premise, and Maclaurin won the prize. In 1728-29 there was a scuffle over whose analytic methods are "better" for mechanics, Newton's or Leibniz's.

The controversy was ostensibly over what gets to be the "true quantity of motion", momentum or vis viva (kinetic energy), with Newton and Leibniz on the opposing sides. While there was some philosophical angle at first it quickly deteriorated into a sideshow to the priority dispute over the invention of calculus, which fueled it, with the substantive part getting lost in all the acrimony.

After 1743 Euler and D'Alambert portrayed the controversy as a debate about words, which became the consesus. Here is Mach's summary: "Investigations of Newton really proved that for free material systems not acted on by forces the Cartesian sum Σmv is a constant, and the investigations of Hyugens showed that also the sum Σmv^2 is a constant... The dispute raised by Leibniz rested, therefore on various misunderstandings. It lasted 57 years till the appearance of D'Alambert's Traite de Dynamique in 1743".

The philosophical side was only tangentially related to energy and momentum. Leibniz criticized mechanistic Cartesian philosophy for not explaining the "source of the vitality" of matter. Newton concurred, but for this very reason to him force should have remained a fundamental concept of mechanics, irreducible to masses and speeds. So he opposed the elevation of vis viva to a metaphysical status favored by Leibniz. More details can be found here.

The starting point of the controversy was Descartes defining "momentum" as mass times speed (not velocity) in the tradition of medieval impetus, and claiming that its total value is conserved. Leibniz gave an example with falling bodies demonstrating that Descartes's "momentum" is not conserved. By that time Huygens already established that the sum of masses times speeds squared is conserved in elastic collisions (he also gave a form of "Newton's" second law), so Leibniz declared that the "true quantity of motion" and called it vis viva. In the meantime Wallis gave the correct definition of momentum as mass times velocity, and stated the conservation law correctly. Newton incorporated that into Principia terming it, you guessed it, "quantity of motion". And so it began.

Partly, the semantic debate was inevitable since the modern definitions of basic mechanical notions weren't established yet. Partly, separate issues got entangled with the original controversy. In 1724 Paris Academy offered a prize for the "best" way of calculating collisions between absolutely hard bodies. Johann Bernoulli's submission stated that... there are no absolutely hard bodies, all collisions are elastic, and, by the way, vis viva is the true quantity of motion. In return Maclaurin suggested calling mv the "force of bodies", and applying Newton's laws to it. Bernoulli's submission was rejected because he rejected the Academy's premise, and Maclaurin won the prize. In 1728-29 there was a scuffle over whose analytic methods are "better" for mechanics, Newton's or Leibniz's.

It started with Descartes defining "momentum" as mass times speed (not velocity) in the tradition of medieval impetus, and claiming that its total value is conserved, so it is the "true quantity of motion". Leibniz gave an example with falling bodies demonstrating that Descartes's "momentum" is not conserved. By that time Huygens already established that the sum of masses times speeds squared is conserved in elastic collisions (along with "Newton's" second law), so Leibniz declared that the "true quantity of motion" and called it vis viva.

  In the meantime Wallis gave the correct definition of velocity as mass times velocity, and stated the conservation law correctly. Newton incorporated that into Principia terming it, you guessed it, "quantity of motion". After that the debate was over what gets to be called "quantity of motion", with many participants positioning themselves along the calculus priority lines. Here is Mach's summary: "Investigations of Newton really proved that for free material systems not acted on by forces the Cartesian sum Σmv is a constant, and the investigations of Hyugens showed that also the sum Σmv^2 is a constant... The dispute raised by Leibniz rested, therefore on various misunderstandings. It lasted 57 years till the appearance of D'Alambert's Traite de Dynamique in 1743".

  Partly, separate issues got entangled with the original controversy. In 1724 Paris Academy offered a prize for the "best" way of calculating collisions between absolutely hard bodies. Johann Bernoulli's submission stated that... there are no absolutely hard bodies, all collisions are elastic, and, by the way, vis viva is the true quantity of motion. Maclaurin suggested calling mv the "force of bodies", and applying Newton's laws to it, he won the prize. In 1728-29 there was a scuffle over whose methods are "better" for mechanics, Newton's or Leibniz's.

Partly, the semantic debate was inevitable since the modern definitions of basic mechanical notions weren't established yet. After 1743 Euler and D'Alambert portrayed the controversy as a debate about words, which it mostly was. The philosophical side was only tangentially related to energy and momentum. Leibniz criticized mechanistic Cartesian philosophy for not explaining the "source of the vitality" of matter. Newton concurred, but to him for this very reason force should have remained a fundamental concept of mechanics, irreducible to masses and speeds. So he opposed the elevation of vis viva to a metaphysical status. More details can be found here.

The controversy was ostensibly over what gets to be the "true quantity of motion", momentum or vis viva (kinetic energy), with Newton and Leibniz on the opposing sides. While there was some philosophical angle at first it quickly deteriorated into a sideshow to the priority dispute over the invention of calculus, which fueled it, with the substantive part getting lost in all the acrimony.

After 1743 Euler and D'Alambert portrayed the controversy as a debate about words, which became the consesus. Here is Mach's summary: "Investigations of Newton really proved that for free material systems not acted on by forces the Cartesian sum Σmv is a constant, and the investigations of Hyugens showed that also the sum Σmv^2 is a constant... The dispute raised by Leibniz rested, therefore on various misunderstandings. It lasted 57 years till the appearance of D'Alambert's Traite de Dynamique in 1743".

The philosophical side was only tangentially related to energy and momentum. Leibniz criticized mechanistic Cartesian philosophy for not explaining the "source of the vitality" of matter. Newton concurred, but for this very reason to him force should have remained a fundamental concept of mechanics, irreducible to masses and speeds. So he opposed the elevation of vis viva to a metaphysical status favored by Leibniz. More details can be found here.

The starting point of the controversy was Descartes defining "momentum" as mass times speed (not velocity) in the tradition of medieval impetus, and claiming that its total value is conserved. Leibniz gave an example with falling bodies demonstrating that Descartes's "momentum" is not conserved. By that time Huygens already established that the sum of masses times speeds squared is conserved in elastic collisions (he also gave a form of "Newton's" second law), so Leibniz declared that the "true quantity of motion" and called it vis viva. In the meantime Wallis gave the correct definition of momentum as mass times velocity, and stated the conservation law correctly. Newton incorporated that into Principia terming it, you guessed it, "quantity of motion". And so it began.

Partly, the semantic debate was inevitable since the modern definitions of basic mechanical notions weren't established yet. Partly, separate issues got entangled with the original controversy. In 1724 Paris Academy offered a prize for the "best" way of calculating collisions between absolutely hard bodies. Johann Bernoulli's submission stated that... there are no absolutely hard bodies, all collisions are elastic, and, by the way, vis viva is the true quantity of motion. In return Maclaurin suggested calling mv the "force of bodies", and applying Newton's laws to it. Bernoulli's submission was rejected because he rejected the Academy's premise, and Maclaurin won the prize. In 1728-29 there was a scuffle over whose analytic methods are "better" for mechanics, Newton's or Leibniz's.